Aircraft guidance in Mathematica

An aircraft flying under the guidance of a nondirectional beacon (a fixed radio transmitter, abbreviated NDB) moves so that its longitudinal axis always points toward the beacon (see Figure 3.19). A pilot sets out toward an NDB from a point at which the wind is at right angles to the initial direction of the aircraft; the wind maintains this direction. Assume that the wind speed and the speed of the aircraft through the air (its “airspeed”) remain constant. (Keep in mind that the latter is different from the aircraft’s speed with respect to the ground.) (a) Locate the flight in the xy-plane, placing the start of the trip at and at . Set up the differential equation describing the aircraft’s path over the ground. Hint: (b) Make an appropriate substitution and solve this equation. [Hint: See Section 2.6.] (c) Use the fact that x 2 and y 0 at t 0 to determine the appropriate value of the arbitrary constant in the solution set. (d) Solve to get y explicitly in terms of x. Write your solution in terms of a hyperbolic function. (e) Let be the ratio of windspeed to airspeed. Using a software package, graph the solutions for the cases 0.1, 0.3, 0.5, and 0.7 all on the same set of axes. Interpret these graphs.